Let $V$ be a vector space over $\mathbb{C}$. Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring $\mathbb{C}[X]$ is arithmetically Cohen-Macaulay (aCM) is the same true for $\mathbb{C}[X^\vee]$?

An example where this is true is when $X$ is $n\times n$ matrices of rank $1$ and $X^\vee$ is $n\times n$ matrices of rank $n-1$.

I guess that it is true in general. Does someone have a reference (or a counter-example)?

Thanks!